As I write this, Hurricane Sandy is bearing down on the east coast of the United States. Mayor Bloomberg has ordered evacuations from various parts of New York City. All over the region people are stocking up on food and other essentials and waiting for Sandy to arrive. And if Sandy doesn’t turn out to be the worst storm ever, will people be relieved or disappointed? Either way there is a lot of money involved. And more importantly, risk of human injury and death. Will the forecasters be blamed for over-predicting?
There are two ways to get this sort of decision wrong. We can do something and find out it was a waste of time, or we can do nothing and wish that we had done something. In the subject of statistics these are known as Type 1 and Type 2 errors. Teaching about Type 1 and Type 2 errors is quite tricky and students often get confused. Does it REALLY matter if they get them around the wrong way? Possibly not, but what really does matter is that students are aware of their existence. We would love to be able to make decisions under certainty, but most decisions involve uncertainty, or risk. We have to choose between the possibility of taking an opportunity and finding out that it was a mistake, and the possibility of turning down an opportunity and missing out on something.
In another recent event, Italian scientists have been convicted of manslaughter for failing to predict a catastrophic earthquake. This has particular resonance in Christchurch as our city has recently been shaken by several large quakes and a multitude of smaller aftershocks. You can see a graph of the Christchurch shakes at this site. In most part the people of Christchurch understand that it is not possible to predict the occurrence of earthquakes. However it seems that the scientists in Italy may have overstated the lack of risk. Just because you can’t accurately predict an earthquake, it doesn’t mean it won’t happen. Here is a link to a story by Nature of the Italian earthquake.
Laura McLay wrote a very interesting post entitled. “what is the optimal false alarm rate for tornado warnings?” . A high rate of false alarms is likened to the “boy who cried wolf”, to whom nobody listens any more. You would think that there is no harm in warning unnecessarily, but in the long term there is potential loss of life because people fail to heed subsequent warnings.
Pure mathematicians tend not to like statistics much as it isn’t exact. It’s a little bit sullied by its contact with the real world. However Operations Research goes a step further into the messy world of reality and evaluates the cost of each type of error. Decisions are often converted into dollar terms within decision analysis. Like it or not, the dollar is the usual measure of worth, even for a human life, though sometimes a measure called “utility” is employed.
Sometimes there is very little cost to a type 2 error. A bank manager refusing to fund a new business is avoiding the risk of a type 1 error, which would result in a loss of money. They then become open to at type 2 error, that they missed out on funding a winner. The balance is very much on the side of avoiding a type 1 error. In terms of choosing a life partner, some people are happy to risk a type 1 error, and marry, while others, hold back, perhaps invoking a type 2 error by missing out on a “soul-mate”. Or it may be that we make this decision under the illusion of certainty and perfect information, and the possible errors do not cross our minds.
Cancer screening is a common illustration of type 1 and type 2 errors. With a type 1 error, we get a false positive and are told we have a cancer when we do not. With type 2, the test fails to detect a cancer. In this example the cost of a type 2 error seems to be much worse than type 1. Surely we would rather know if we have cancer? However in the case of prostate cancer, a type 1 error can lead to awful side-effects from unnecessary tests. Conversely a large number of men die from other causes, happily unaware that they have early stages of prostate cancer.
The point is that there is no easy answer when making such decisions.
I have found the following helpful when teaching about type 1 and type 2 errors in statistics. Think first about the action that was taken. If the null hypothesis was rejected, we have said that there is an effect. After rejecting the null only two outcomes are possible. We have made the correct decision, or we have made a type 1 error. Conversely if we do not reject the null hypothesis, and do nothing, we have either been correct or made a type 2 error. You cannot make a type 1 error and a type 2 error in the same decision.
Or another way of looking at it is:
Students may wonder why we have to have any kind of error. Can we not do something to remove error? In some cases we can – we can spend more money and take a larger sample, thus reducing the likelihood of error. However, that too has its cost. The three costs are important aspects of decision-making, and helping students to understand this will help them to make and understand decisions.